On the uniqueness of solutions to relations based
on products and mixed sums of random variables

Abstract. We discuss uniqueness of solutions to \(Z \stackrel{d}{=} XY\) and \(Z \stackrel{d}{=} U_1 X_1 +U_2 X_2 +\cdots +U_l X_l\), where the random variables are independent and \(Z, X_1, \ldots , X_l\) are identically distributed. We also discuss symmetry of products and quotients of independent random variables.